Reader’s guide to the Articulated Course · episode 6 / 8 · 4 min 20 s
The ten companion objects
Ten objects, from the circle to the torus, that return from volume to volume.
In this episode
Ten mathematical objects run through the collection like guiding threads: the circle, the integers, the extension of ℚ by √2, an elliptic curve, the group GL₂, the zeta function, the space L², the ring K[X], the symmetric group 𝔖₃ and the torus. Sixth episode of the reader’s guide: how each one returns from volume to volume with new depth.
In the book: “To the reader” and conclusion of volume I; forewords of volumes II, III and IV.
Read the transcript
An example that does not disappear
In most courses, an example is used once, then disappears. Here, ten mathematical objects come back from volume to volume, each time deeper. The foreword calls them guiding threads. Articulated Course in Mathematics. Reader’s guide, episode six: the ten companion objects.
The ten objects
Here are the ten. The circle. The integers. The extension of Q by root two. An elliptic curve. The group G L two. The zeta function. The space L two. The ring of polynomials. The symmetric group on three elements. The torus. From trigonometric calculation to the Langlands dual group, from the harmonic oscillator to Montonen-Olive duality, each one reappears in every cycle with new depth. When one of them appears, the text says so.
The circle, volume by volume
Let’s follow the circle. In Elements, it is first the home of angles, then the support of cosine and sine, then the theatre of complex multiplication. In Foundations, it becomes the group of complex numbers of modulus one. In Graduate Mathematics, it is a differentiable manifold, and a Lie group. It is always the same circle. Each volume gives it one more role.
In Foundations
In Foundations, four of the ten are treated explicitly. The integers turn out to be Euclidean, principal, factorial. The symmetric group on three elements appears as the smallest non-abelian group. The polynomial ring receives its Euclidean division and becomes the model principal ring. The circle you have just seen. The other six, the elliptic curve, the group G L two, the zeta function, the extension of Q by root two, the space L two and the torus, stay in the background. They wait for the later volumes.
In Graduate Mathematics
In Graduate Mathematics, they become laboratories where each new theory is tested. The integers: a Euclidean ring whose ideals containing n Z describe the subgroups of Z mod n Z. The extension of Q by root two: a number field whose Galois group is Z mod two Z. The torus: the quotient of R two by Z two, natural ground for Fourier analysis.
In Correspondences
In Correspondences, familiar objects return with new functions. In the extension of Q by root two, integers, units and places bring algebra and arithmetic together. An elliptic curve carries rational points, Tate modules and an L-function. The group G L two lets us compare a representation, an automorphic form and a modification of a bundle. These returns make the continuity of the collection tangible.
The pedagogical companions
Alongside the ten, Foundations calls on other, more modest, pedagogical companions: the plane R two, the polynomial X squared plus one, the exponential, the prime numbers, the finite field F p. They structure the progression of the chapters, and are often called on as examples.
To remember
To remember. Ten objects run through the collection, like guiding threads. Each volume gives them one more role. They are not examples: they are companions. The list of the ten is in the foreword of volume one, in the free extract. The link is in the description. Next episode: the three companion books.
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